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I've been reading some "introduction to categories" type materials and have been impressed with the all-encompassing nature, but the skeptic in me wonders: is there any mathematical object that categories can't describe?

To be quite specific, I'd be interested any of these:

a.) Objects that can be described by categories that have properties that can't.

b.) Category equivalents of set-theoretic type limits, like how "the set of all sets" causes problems.

c.) Some type of mathematics so pathological it foils, say, associativity. It doesn't need to be a mathematics that's useful in any sense, just one designed specifically to be impossible to describe with categories.

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To properly describe objects like categories themselves, you should really work with 2-categories. Probably not the kind of answer you were looking for, though. –  Reid Barton Oct 22 '09 at 17:27
    
Well, close: is there some sort of Russell's Paradox for categories? –  Jason Dyer Oct 22 '09 at 17:35
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I'm still not sure what you mean by "describe." If you think sets are an adequate description of mathematical objects, then the category of sets is also an adequate description of mathematical objects. –  Qiaochu Yuan Oct 22 '09 at 18:11
    
There are standard maneuvers to get around size issues, which let you say most of the things you want to say while avoiding paradoxes. See Scott's answer for some references. –  Reid Barton Oct 22 '09 at 19:00
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7 Answers

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I don't quite understand your question, but if you're asking whether category theorists should worry about set-theoretic problems the answer seems to be "sometimes". I'm not an expert in this area, but it seems that people tend to avoid universal constructions like limits over large diagrams, and in other cases, people assume the existence of strongly inaccessible cardinals. This seems to avoid standard contradictions, but I must confess that I've never checked such arguments.

I don't know many references for this question. Lurie discusses some constructions in section 5.4 of Higher topos theory.

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See also section 1.2.5 "Set-Theoretic Technicalities" for a related discussion. –  Reid Barton Oct 22 '09 at 18:11
    
This answer was closest to what I meant, but the others are extremely helpful as well, thank you. –  Jason Dyer Oct 23 '09 at 1:07
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Category theory just provides a language. Sometimes it provides really useful descriptions and metaphors and theorems, but most theorems can't be proved using category theory (even if they can be nicely described in that language). The English language can be used to describe any mathematical object you can think of, and most proofs today are written in English, but that doesn't mean that somehow English has some deep mathematical significance.

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I never understand this phrase "Category theory is just a language." What is, say, differential geometry if not a language? In fact, when diffential geoemtry was introduced into physics, that's what many physicists asked. Mayne of them still say today: "Oh, I don't really know what a 1-form is. Just some language that mathematicians use when they see a lower index". In this example, few people doubt that the choice of the right kind of language goes a long way towards understanding phenomena. –  Urs Schreiber Nov 6 '09 at 7:57
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Someone (Halmos?) said that category theory can only describe the most trivial parts of any subject, and that this a valuable service because it's good to clearly mark which parts are trivial.

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See also ncatlab.org/nlab/show/nPOV#RoleOfnPOV. –  Todd Trimble Sep 11 '10 at 18:25
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There's an interesting variant of your question, which may perhaps have been included in the first part of it, as to whether there are parts of mathematics where categories have little traction, and whether this is a necessary state of affairs. I was driving at this distinction when I posed a question to Terry Tao which he answered here.

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I don't know if this is the kind of answer you're looking for, but category-theoretic language cannot be used to define some properties you may be very interested in. For example, if you're working in a category where the objects have underlying sets, then the notion of a morphism being surjective is not a categorical one. You can prove that in the category of sets, surjectivity is equivalent to the map being an epimorphism, but this is not true in the category of topological spaces (where epimorphism corresponds to the image being dense).

So you can't do everything with abstract nonsense. Sometimes, you actually have to prove things by using special facts about the specific category you're working with. But those results can often be interpreted as "rules about how things work in the category". For example, you might prove that if two morphisms have a certain property P, then when you compose them, you still get something with property P. Or maybe you can show that "property P is stable under base change" (a common one in algebraic geometry). To prove these things, you have to "get your hands dirty," but once you have those results, you can use them in your category-theoretic arguments.

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@ paragraph 1: of course, the notion of 'in a category C, objects have underlying sets' means I have a certain functor F: C --> Set, and the notion of surjectivity of a morhpism f in C is then whether or not F(f) is a categorical epimorphism in Set. –  Harold Williams Oct 22 '09 at 20:33
    
@Harold: The point is that by appealing to the forgetful functor, you're "looking inside the objects" rather than just using the structure of the category. (Actually, you can get the notion of surjectivity without appealing to what the objects actually are. A morphism X-->Y is surjective if the corresponding map on hom sets Hom(*,X)-->Hom(*,Y) is surjective, where * is the terminal object in Top. So you're right that this was a bad example.) –  Anton Geraschenko Oct 22 '09 at 21:29
    
Because your forgetful functor is Hom(,-). This is a good forgetful functor, and up to canonical isomorphism a canonical one if your category has a terminal object. When I say "underlying set", I demand that my forgetful functor be faithful. Is it true that Hom(,-) is always faithful if * is the terminal object? –  Theo Johnson-Freyd Oct 22 '09 at 21:55
    
Er, in the future, how do I escape an asterisk, to make it not italicize everything? In my comment above, the italic section should be roman, with a * at the front and one at the back. –  Theo Johnson-Freyd Oct 22 '09 at 21:56
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backslash-asterisk is how you escape an asterisk. The functor Hom(*,-) is not very often faithful, e.g., take any pointed category, where the final object is also initial, or even the category of sheaves on a space, where this is the global sections functor. –  Reid Barton Nov 5 '09 at 15:46
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From the point of view of someone who uses mainly analytic inequalities applied to areas of geometric analysis and PDE's, the more natural question is "why is category theory useful at all?"

Category theory appears to be a resounding success in the more algebraic areas of mathematics, but has had far less impact in most areas of analysis and differential geometry. Why this is so I will leave to people who understand this much better than me.

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You can also ask, why the abstract definition of a group is useful at all. People where working with fuzzy concepts of groups for quite a long time, so you can do mathematics without axiomatizing (is this a proper word?) the objects you are talking about. But once you do this, you might see properties that you just couldn't discover in explicit examples. It's like verifying each time for an explicit group that there exists a unit element. –  user717 Oct 22 '09 at 21:12
    
(a unique unit element of course... :) –  user717 Oct 22 '09 at 21:17
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Catgegory theory becomes important as soon as you feel yourself touching the conceptual infrastructure of your topic. The reason why it is so highly developed in algebraic geometry is that when you start working on the perspective that spaces should be modeled by algebras, you very soon find yourself having to think closely about what "space" actually should mean. The "problem" with differential geometry is that manifolds are more flexible than varieties, so people can fake it much longer and pretend that everything is a manifold. But this is changing now a bit also in diff geometry. –  Urs Schreiber Nov 6 '09 at 8:03
    
What exactly do you mean by 'fake'? –  Martin Dec 5 '09 at 2:19
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Well, if you don't impose additional conditions, then of course even the category of all sets does not exist. To define such a category you either have to restrict to, say, sets of bounded rank or you have to take a set theory with a notion of classes for example. The next step is the category of categories or functor categories. Here the situation is even worse (but similar). There are a lot of constraints imposed on category theory by set theory. Therefore people start to think about using categories and not set theory as foundation of mathematics. As far as I know, there is now adequate solution to these problems.

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